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A274106 Triangle read by rows: T(n,k) = total number of configurations of k nonattacking bishops on the white squares of an n X n chessboard (0 <= k <= n-1+[n=0]).

%I A274106 #50 Feb 16 2025 08:33:36
%S A274106 1,1,1,2,1,4,2,1,8,14,4,1,12,38,32,4,1,18,98,184,100,8,1,24,188,576,
%T A274106 652,208,8,1,32,356,1704,3532,2816,632,16,1,40,580,3840,12052,16944,
%U A274106 9080,1280,16,1,50,940,8480,38932,89256,93800,37600,3856,32,1,60,1390,16000,98292,322848,540080,412800,116656,7744,32
%N A274106 Triangle read by rows: T(n,k) = total number of configurations of k nonattacking bishops on the white squares of an n X n chessboard (0 <= k <= n-1+[n=0]).
%C A274106 From _Eder G. Santos_, Dec 16 2024: (Start)
%C A274106 The sequence counts every possible nonattacking configuration of k bishops on the white squares of an n X n chess board.
%C A274106 It is assumed that the n X n chess board has a black square in the upper left corner.
%C A274106 (End)
%H A274106 Irving Kaplansky and John Riordan, <a href="http://projecteuclid.org/euclid.dmj/1077473616">The problem of the rooks and its applications</a>, Duke Mathematical Journal 13.2 (1946): 259-268. See Section 9.
%H A274106 Irving Kaplansky and John Riordan, <a href="/A274105/a274105.pdf">The problem of the rooks and its applications</a>, in Combinatorics, Duke Mathematical Journal, 13.2 (1946): 259-268. See Section 9. [Annotated scanned copy]
%H A274106 J. Perott, <a href="https://doi.org/10.24033/bsmf.267">Sur le problème des fous</a>, Bulletin de la S. M. F., tome 11 (1883), pp. 173-186.
%H A274106 Eder G. Santos, <a href="https://arxiv.org/abs/2411.16492">Counting non-attacking chess pieces placements: Bishops and Anassas</a>. arXiv:2411.16492 [math.CO], 2024. (considered as black board).
%H A274106 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/WhiteBishopGraph.html">White Bishop Graph</a>.
%F A274106 From _Eder G. Santos_, Dec 01 2024: (Start)
%F A274106 T(n,k) = Sum_{j=0..k} binomial(floor(n/2),j) * Stirling2(n-j,n-k).
%F A274106 T(n,k) = T(n-1,k) + (n-k+1-A000035(n)) * T(n-1,k-1), T(n,0) = 1, T(0,k) = delta(k,0). (End)
%e A274106 Triangle begins:
%e A274106   1;
%e A274106   1;
%e A274106   1,  2;
%e A274106   1,  4,    2;
%e A274106   1,  8,   14,     4;
%e A274106   1, 12,   38,    32,     4;
%e A274106   1, 18,   98,   184,   100,      8;
%e A274106   1, 24,  188,   576,   652,    208,      8;
%e A274106   1, 32,  356,  1704,  3532,   2816,    632,     16;
%e A274106   1, 40,  580,  3840, 12052,  16944,   9080,   1280,     16;
%e A274106   1, 50,  940,  8480, 38932,  89256,  93800,  37600,   3856,   32;
%e A274106   1, 60, 1390, 16000, 98292, 322848, 540080, 412800, 116656, 7744, 32;
%e A274106   ...
%e A274106 From _Eder G. Santos_, Dec 16 2024: (Start)
%e A274106 For example, for n = 3, k = 2, the T(3,2) = 2 nonattacking configurations are:
%e A274106   +---+---+---+   +---+---+---+
%e A274106   |   | B |   |   |   |   |   |
%e A274106   +---+---+---+   +---+---+---+
%e A274106   |   |   |   | , | B |   | B |
%e A274106   +---+---+---+   +---+---+---+
%e A274106   |   | B |   |   |   |   |   |
%e A274106   +---+---+---+   +---+---+---+
%e A274106 (End)
%p A274106 with(combinat): with(gfun):
%p A274106 T := n -> add(stirling2(n+1,n+1-k)*x^k, k=0..n):
%p A274106 # bishops on white squares
%p A274106 bish := proc(n) local m,k,i,j,t1,t2; global T;
%p A274106     if n=0 then return [1] fi;
%p A274106     if (n mod 2) = 0 then m:=n/2;
%p A274106         t1:=add(binomial(m,k)*T(2*m-1-k)*x^k, k=0..m);
%p A274106     else
%p A274106         m:=(n-1)/2;
%p A274106         t1:=add(binomial(m,k)*T(2*m-k)*x^k, k=0..m+1);
%p A274106     fi;
%p A274106     seriestolist(series(t1,x,2*n+1));
%p A274106 end:
%p A274106 for n from 0 to 12 do lprint(bish(n)); od:
%t A274106 T[n_] := Sum[StirlingS2[n+1, n+1-k]*x^k, {k, 0, n}];
%t A274106 bish[n_] := Module[{m, t1, t2}, If[Mod[n, 2] == 0,
%t A274106    m = n/2;     t1 = Sum[Binomial[m, k]*T[2*m-1-k]*x^k, {k, 0, m}],
%t A274106    m = (n-1)/2; t1 = Sum[Binomial[m, k]*T[2*m - k]*x^k, {k, 0, m+1}]];
%t A274106 CoefficientList[t1 + O[x]^(2*n+1), x]];
%t A274106 Table[bish[n], {n, 1, 12}] // Flatten (* _Jean-François Alcover_, Jul 25 2022, after Maple code *)
%o A274106 (SageMath) def stirling2_negativek(n, k):
%o A274106   if k < 0: return 0
%o A274106   else: return stirling_number2(n, k)
%o A274106 print([sum([binomial(floor(n/2), j)*stirling2_negativek(n-j, n-k) for j in [0..k]]) for n in [0..10] for k in [0..n-1+kronecker_delta(n,0)]]) # _Eder G. Santos_, Dec 01 2024
%Y A274106 Columns k=0-1 give: A000012, A007590.
%Y A274106 Alternate rows give A088960.
%Y A274106 Row sums are A216078(n+1).
%Y A274106 T(2n,n) gives A191236.
%Y A274106 T(2n+1,n) gives A217900(n+1).
%Y A274106 T(n+1,n) gives A060546.
%Y A274106 Cf. A274105 (black squares), A288182, A201862, A002465.
%K A274106 nonn,tabf
%O A274106 0,4
%A A274106 _N. J. A. Sloane_, Jun 14 2016
%E A274106 T(0,0) prepended by _Eder G. Santos_, Dec 01 2024